A New Primal-Dual Weak Galerkin Method for Elliptic Interface Problems with Low Regularity Assumptions

TL;DR

This paper presents a novel primal-dual weak Galerkin finite element method tailored for elliptic interface problems with very low regularity solutions, demonstrating stability, optimal error estimates, and verified efficiency through numerical experiments.

Contribution

It introduces a new PDWG method that handles ultra-low regularity assumptions, providing stability and optimal error estimates for elliptic interface problems.

Findings

Method achieves optimal error estimates in Sobolev norms.

Numerical experiments confirm the method's efficiency and accuracy.

Applicable under low regularity conditions for solutions and interfaces.

Abstract

This article introduces a new primal-dual weak Galerkin (PDWG) finite element method for second order elliptic interface problems with ultra-low regularity assumptions on the exact solution and the interface and boundary data. It is proved that the PDWG method is stable and accurate with optimal order of error estimates in discrete and Sobolev norms. In particular, the error estimates are derived under the low regularity assumption of $u\in H^{\delta}(\Omega)$ for $\delta > \frac12$ for the exact solution $u$. Extensive numerical experiments are conducted to provide numerical solutions that verify the efficiency and accuracy of the new PDWG method.